[Kuroshima]

It's really a matter of different flavors:

Strength gives you a better rounded character. If you buy full strength, you end up with a character that can strike hard (Striking ST), take a blow (HP) and lift heavy things (Lifting ST). Even if you just buy Striking ST, the character can do kicks, punches, elbow strikes, etc... and use weapons. If the character has an attack that is distinctively stronger than the rest, you just buy Striking ST with One Attack Only, representing that his Crane Kick is awesome. Skill-wise, you get the benefit of having loads of options and techniques, but unless you invest in a non-cost effective number of techniques, you can do all the tricks in your arsenal, you just do some of them better (For example, the character has Striking ST with One Attack Only for his crane kick, and then he has Karate plus the Lethal Kick technique). ST also gets loads of "force multipliers", in that you can get bonuses per dice from skill (karate, brawling, boxing, or weapon master if using a weapon), use cinematic skills to multiply your ST (Power Blow/Flying Leap), or add extra effects to your attacks (Pressure Points/Secrets, Imbuements). These force multipliers more often than not cost a flat amount of points, independent of your ST, but as their name suggest often provide greater benefits the higher your ST is (thus multiplying it's effect).

Innate Attack on the other hand represent distinct attacks, and there's no cross-benefits. You might have a Stone Breaking Punch with armor divisor, but you can't really hit hard with a kick or a weapon. Your attack will deal crushing damage, and you can't make it do Impaling damage without using the stunt rules. Sure, you could build a dozen alternate attacks representing strong strikes (crushing attack with loads of dice), fast strikes that can not be defensed against (crushing attack with cosmic, no active defense), strikes charged with body energy that pierce through inanimate matter like it wasn't there (crushing attack with armor divisor or cosmic, irresistible attack), shoves to push (Innate Attack with double knockback and some or all levels limited with no wounding), strikes that deal damage as blades (cutting or impaling innate attacks) pressure point strikes that inflict various effects on your target (various afflictions), etc or you could use the power stunts rules to go beyond what's written in your character sheet, but trying to replicate all the tricks of the ST fighter will not be cheap, and will result in a lot of complex traits in your character sheet. Also, there will be a big disparity between what you can do with your special attacks, and what you can do with mundane equipment.

Using Strength is more "realistic", and so it's appropriate for more kinds of games, while using Innate Attacks can be more point efficient*.

*Where it stops being point efficient is when you try to cross over, as in having both a slew of innate attacks plus striking ST, and then you also want HP, and lifting ST. Of course, you could take the Striking ST as another ability of the alternate abilities group, but since it's not an attack, you would need to take a ready maneuver to switch between innate attacks and ST.